CARESS Working Paper 98-10
Ergodicity and Clustering in Opinion Formation¤y.
Antonella Ianniz
University of Southampton (U.K.),
University of Pennsylvania (U.S.A.)
Valentina Corradi
University of Pennsylvania (U.S.A.)
April, 27, 1998
Abstract
We study a simple model of pre-electoral opinion formation that posits that
interaction between neighbouring voters leads to bandwagons in the dynamics
of the individual process, as well as in that of the aggregate process. We show
that in di¤erent speci…cations of the model, there is a tendency for the process
to show consensus, i.e. to approach a con…guration of homogeneous support
for one candidate, out of the two who run the electoral campaign. We point
out that the process displays the feature that, after long time spans, a sequence
of states occur which, when viewed locally, remain almost stationary and are
characterized by large clusters of individuals of the same opinion.
¤
y
JEL: D72, D82, C44. Keywords: Ergodicity, Clustering, Local interaction, Voter model.
We have bene…ted from fruitful comments from L. Anderlini, D. Cass, S. Coate, N. Persico, M.O.
Ravn, L. Samuelson, and all seminar participants at the University of Pennsylvania. The …rst author
wishes to thank the University of Southampton for granting a research leave and the University of
Pennsylvania for the generous hospitality.
z
Address for Correspondence: A. Ianni, Department of Economics, University of Southampton, Southampton SO14 2AY, U.K.. e-mails: a.ianni@soton.ac.uk, aianni@econ.sas.upenn.edu
1
Public opinion, if we wish to see it as it is, should
be regarded as an organic process, and not merely as a
state of agreement about some question of the day.
Cooley (1918) (p. 378)
1
Introduction
Almost a century has passed since Cooley’s words and though scholars, observers and
analysts do not seem to agree as to what public opinion actually is, there seems to be a
widespread consensus that it should be studied as an “interactive, multidimensional,
continuously changing phenomenon whose diverse aspects form causally interrelated
patternings” (Crespi (1997), p. ix1 ). Two particular aspects are often emphasized in
the sociological literature. The …rst is the fact that individuals faced with di¤erent
choices as to whom - or what - to support show a tendency to be in‡uenced by the
opinion of some collective majority (mutual awareness, as de…ned in Crespi (1997)).
The second is that environmental conditions that are speci…c to each agent seem
to matter in determining the outcome of individual choices (situational correlates of
opinion, as in Crespi (1997)). These features of the public opinion process seem to
be well documented in terms of experimental and empirical evidence.
Issues related to the process of public opinion formation are not tangential to
economic theory. Public opinion plays a key role in shaping animal spirits, expectations, voting decisions, patterns of consumer and producer behaviour, as well as
dynamics of adoptions of di¤erent technologies and innovation. In a nutshell, as common sense has it, public opinion may and does a¤ect daily life, as well as business
cycles. Indeed a relevant body of modern economic theory, in particular in the …eld
of Finance (Banerjee (1992), Birkhandani, Hirshleifer and Welch (1992), Lee (1993),
Devenow and Welch (1996) among others) and Industrial Organization (Farrel and
Saloner (1985), Katz and Shapiro (1985), Kandori and Rob (1998) among others),
has often dealt with issues arising from network externalities and band-wagon e¤ects,
where the formalizations explicitly postulate a dependence of the level of utility or of
pro…t of a single individual on the proportion of other individuals who think and act
in an analog fashion. From the point of view of pure theory a great deal of attention
2
has been devoted to the analysis of simple models that admit multiple equilibria and
in particular coordination games, where the underlying incentive structure supports
herding as equilibrium behaviour (Vega-Redondo (1995) and therein references).
This paper formalizes and studies a simple process of private and public opinion
formation. The main aim is to account for an individual process by which each
agent forms an opinion, as well as an aggregate process that shapes (and in many
respects de…nes) public opinion in its collective dimension. As a metaphor that helps
us describing these processes, we think of a model of pre-electoral opinion formation,
where individuals repeatedly form their own opinion as to which, out of two, candidate
to vote for, at the time when elections come.
As there are very many voters in the population, voting decisions are almost by
de…nition deprived of any strategic element and each voter sees the future electoral
outcome as an unknown state of the world. Nevertheless, in order to account for an
explicitly interactive element, we postulate that preference are such that each voter
has an incentive to conform to what (s)he perceives to be the winning side of the
elections. The motivation we have in mind is in terms of band-wagon e¤ect (voters
favour the party that is doing well in the polls), or projection e¤ect (voters tend
to project their intended vote onto their election outcome expectations). Since the
theoretical studies of Simon (1954) and Baumol (1957), empirical work carried out in
the UK and in the USA seems to provide evidence for these hypothesis (see McAlister
and Studlar (1991), Zuckerman A.S., Valentino and Zuckerman E.W. (1994) and
therein references).
We formally think in terms of a side-payment that each voter will receive if (s)he
votes for the candidate who wins the elections. This raises an incentive for an individual to gather some information on the current state of public opinion, as this
would determine, through a simple majority voting, the electoral outcome. To capture the fact that choices are often determined by the features of the environment
where interaction takes place, we postulate that each voter can only observe the opinions adopted within the small subset of her or his neighbours, colleagues, friends or
relatives, and for modeling purposes we endow each voter with a speci…c location on
an appropriately characterized topological structure. Models of interactive behaviour
with a local connotation have been extensively analyzed in the modern literature on
3
learning and evolution (Blume (1993), Ellison (1993) and (1995), Anderlini and Ianni
(1996), Morris (1997a) and (1997b), Eshel, Samuelson and Shaked (1998) and Ianni
(1998) among others). We see this process of local interaction as being a minor component of each voter’s daily life; as such it takes place repeatedly over time and it
may lead to di¤erent decisions.
We refer to the process of public opinion formation as to the dynamic process
generated by the aggregate of all the individual decisions, and we provide a stochastic
formulation. We are interested in analyzing the properties of the time evolution of
this process. In particular, we address two complementary issues. The …rst relates
to the asymptotic properties of the dynamics. We ask what is likely to happen
after many time periods have elapsed, and whether we are able to provide unique
predictions as to the limit outcome of the process of public opinion formation. The
second focuses on the behaviour of the process along the dynamics and relies on the
explicit characterization of the process of cluster formation.
We believe that the analysis of the rate at which local areas of consensus grow over
time is important in order to understand the dynamics of the process towards its long
run behaviour. This feature is not peculiar to this model, but rather it is a common
feature of many economic settings where multiple equilibria may arise. Indeed, it is
a stylized fact that several economic and social variables show a high degree of local
homogeneity and persistent cross-sectional variance, that is only partly explained
by fundamental di¤erences in economic conditions. This is the case, for example, for
crime (where what is puzzling is not the overall level of criminal activity, but rather its
high variance across time and space), or the persistence of income inequalities (ghetto
formation and poverty traps), or the co-existence of di¤erent, maybe rival, techniques
within an industry of identical …rms (where standard economic theory would view
the adoption of new technologies as essentially monotonic), or phenomena of price
dispersion and tax dispersion (even in the absence of heterogeneities among agents).
Last, but not least, a quick glance at the distribution of votes over geographical
areas show large areas of consensus, certainly in the UK, in the USA and in many
other European countries. The methodology we use allows to characterize situations
where di¤erent opinions co-exist in the population in terms of clusters that are almost
stationary, i.e. that vary very slowly over time. Hence, although these con…gurations
4
cannot be observed in any steady state of the process we study, still they can be a
persistent feature of the dynamics along its evolution2 .
Although the aim of the paper and the questions we address are di¤erent, this
paper also contributes to the recent literature on learning and evolution in interactive settings in two respects. First, the speci…cation of the process of private opinion
formation we model (based on Bayesian updating, given a sample of observations)
produces a dynamics entirely analog to the speci…cation of noisy best-reply dynamics
used for example in Blume (1993), McKelvey and Palfrey (1995) and Ianni (1998).
This paper departs from those speci…cations in that we do not postulate any mistake
on the part of individuals, as the probabilistic component that drives the dynamics
stems entirely from the fact that the information available at the time when choices
are to be made is limited. Second, the aggregate dynamics we study could be applied
to an explicitly interactive setting, where randomly drawn couples of players repeatedly play a one-shot (2-by-2) coordination game. If this line were to be pursued,
the results we obtain here (namely Theorem 2 and Theorem 3) would complement
what is already known in the literature as to the asymptotic properties of myopic
best-reply dynamics, with more information as to the time evolution of the process
towards its steady states.
The paper is organized as follows. Section ?? describes the details of the process
of private and public opinion formation. The individual process is formalized in terms
of an estimate of the current public opinion, on the basis of which a voter forms her
or his private opinion. The collective process relies on two main elements. First, it is
assumed that opinions are formed repeatedly over time in a sequential manner (where
only one voter at a time can revise or formulate an opinion). Second, as observations
consists of other voters’ opinions, the distribution from which observations are drawn
at random is endogeneized in terms of a simple statistic of the voter’s neighbourhood. Section ?? analyses the properties of the dynamic process of public opinion
formation. As anticipated, the study relies on the characterization of the long-run
properties of the dynamics, as well as that of its short-term (or …nite time) features.
The results show that these two aspects are complementary and provide a better
understanding of the process itself. Although decisions are highly decentralized, as
the modelled incentive structure is such that each voter has an incentive to conform
5
to what (s)he perceives as being the current collective opinion, the aggregate process
displays an analog feature. By pursuing a space-time analysis (i.e. by relating the
two dimensions, time and space, over which our process is de…ned) we study the
process of cluster formation. In Section ?? we provide some heuristic considerations
as to the implications that these …ndings would have within a more general model
that allows for strategic behaviour on the part of the two candidates. Finally Section
?? concludes and the Appendix contains the technical proofs, as well as a Remark
on the speci…c characterization we use.
2
The model
The model formalizes in a simple way the process of pre-electoral public opinion
formation. Elections are going to be held at a future date. Two candidates, A and
B, run the elections and the winner will be decided through simple majority voting.
To focus the model on the behaviour of the public, we disregard completely any
strategic element on the part of the candidates. For the purposes of our analysis,
each of them has some well de…ned electoral plan, the implementation of which will
a¤ect each voter’s utility, after the elections are run and the winner is decided.
In the model there are countably many identical voters3 that formulate their
opinion as to which candidate to support when the elections will be held. Voters
behave in an identical manner, though, as we shall see, asymmetries might arise due
to di¤erences in the information they possess. The next two sessions describe the
process of opinion formation on the part of a single voter and the process of public
opinion formation respectively.
2.1
Private Opinion Formation
As there are many voters in the population, voting decisions on the part of each
single voter are almost by de…nition deprived of any strategic content. Only the
pivotal voter will eventually determine the outcome of the elections and, for each
voter, the probability of being pivotal is negligible4 . Hence, undergraduate microeconomics textbook wisdom has it, a rational voter does not exist5 . The model we
formalize takes into account the fact that each voter cannot marginally determine the
6
electoral outcome, but still accounts for speci…c preferences over the two candidates.
In particular, we shall think of each possible electoral outcome as an unknown state
of nature for the single voter. Preferences, formalized by a utility function, depend
on the state of nature, and expected utility considerations determine the process of
opinion formation, and ultimately, the outcome of the elections.
We are going to describe the way in which, given current public opinion, a voter
formulates his or her own. For the voter, the outcome of this process will be an
opinion, a or b, which would correspond to a vote (for A or for B respectively) if
elections were to be held at the same point in time. Voters may be forgetful and go
through this process repeatedly in their electoral life.
Ingredients of this process are: two exogenous states of nature, labelled A and B
corresponding to the event “candidate A wins the elections” and “candidate B wins
the elections”, and a utility function that depends on the outcome of the election and
on the vote chosen. The idea we want to pursue is that, although the outcome of the
elections is exogenous to the voter, utility depends on the vote itself. The simplest
way to formalize this is to think in terms of a side payments, denoted by "A > 0 and
"B > 0, that a voter gets if (s)he has voted for candidate A or B respectively and if
that candidate wins the elections. We further postulate that the utility function is
quasi-linear in this latter argument:
A wins
B wins
vote a
U(A) + "A
U(B)
vote b
U(A)
U(B) + "B
Given any probability distribution, P , over the state space fA wins; B winsg; it
is easy to notice that EP U(a) > EP U(b) if Pr(A wins) > "B ("A + "B )¡1 ´ ". As, in
order to win the elections, candidate A must have the support of at least half of the
population, if we let º A 2 [0; 1] denote the fraction of the electorate who is currently
supporting candidate A, the above inequality can be restated as Pr(º A > 1=2) > ".
If, whenever forming an opinion, the voter knew what was the current public
opinion (i.e. if (s)he knew exactly º A ), then (s)he would form an opinion consistently
with the above inequality. However, we assume that this information is not readily
available, and the need for some inference on the part of the voter arises. We formalize
this process of inference as follows.
7
We assume the voter has ‡at priors over º A 2 [0; 1] which is the fraction of
voters in the population who currently support candidate A, and updates this priors
after having observed a sample of observations. We take priors to be given by a
Beta distribution with equal parameters, Be(1; 1). Each observation consists of a
randomly chosen other voter in the population, the opinion of whom is observed.
Each observation comes from a Binomial distribution, Bi(p), where 0 · p · 1 is the
parameter and all observations are i.i.d.. We denote the density of the probability
distribution that generates observations by fBi (r j n; p) where r is the number of
opinions a in a sample of n observations.
If a voter has observed a sample of r opinions a in a sample of n observations,
_ with density
then (s)he updates her prior Be(1; 1) to the posterior Be(1+r; 1+n¡r),
fBe (z j 1 + r; 1 + n ¡ r) and mean (1 + r)(n + 2)¡1 . Given this posterior, the voter
would choose opinion a if the PrfBe [z > 1=2 j r; n] > "; opinion b otherwise. As
observations come from the above distribution, with parameter p, we can calculate
the probability of the voter choosing opinion a, given n observations as:
Z1
n µ ¶
X
n r
(n + 1)!
n¡r
z r (1 ¡ z)n¡r dz > "g)
Pr[a j n; p; "] =
p (1 ¡ p)
1(f
r!(n ¡ r)!
r
r=0
(1)
1=2
where 1(f¢g) is an indicator function that takes value of one whenever f¢g is true.
This quantity depends in a non trivial way on n (the number of observations), on
p (the parameter that determines the probability of observing an a) and on " (the
threshold value de…ned by the voter’s preferences).
In the Sections that follow we shall formalize a dynamic process of public opinion
formation where in…nitely many voters repeatedly form their opinion, in the way
we set out in this Section, in a setting where information is highly decentralized.
This will naturally introduce a speci…c form of asymmetry among voters: although
they are identical in terms of preferences, as well as in the way they gather and
process information, what will determine their information set (i.e. p in the above
formulation) will be voter speci…c and correlated across di¤erent voters. In order to
achieve this aim, we proceed as follows.
First, we take the parameter " to be a half. This practically means that the side
payments a voter would receive by the winning candidate are identical (though the
8
utility achievable in each state of the world can of course be di¤erent) and creates an
incentive for the voter to form an opinion in favor of the candidate that is supported
by the simple majority of the electorate6 . Furthermore, in order to focus on two
parameters (instead of three, as in (??)) we assume that each voter, given a sample
of observations, draws one realization at random from the probability distribution
de…ned by her updated posterior. Hence, the probability with which (s)he will choose
opinion a is given by PrfBe [z > 1=2 j r; n]. In this case the r.h.s of (??) can be re-
written as:
Z1
n µ ¶
X
n r
n¡r (n + 1)!
z r (1 ¡ z)n¡r dz
Pr[a j n; p] =
p (1 ¡ p)
r!(n ¡ r)!
r
r=0
(2)
1=2
It is important to notice that, unlike (??), (??) formalizes a process that is not
entirely consistent with expected utility maximization, in that the behavior of the
voter, given a sample of observations, is only de…ned probabilistically. The motivation
we provide for (??) relies on the considerations that follow.
Note that (??) depends on n (number of observations) and p (the parameter that
determines the probability of observing an a). In order to understand the behaviour of
this probability, we study its dependence from each of the two arguments separately.
For a …xed n = n the above probability is a continuous function of p 2 (0; 1),
and we denote it by Pr[a j n; p]. It is not di¢cult to show that this is increasing
in p, formalizing the fact that the more likely observations as are, the higher is the
probability that the voter will adopt opinion a.
To see the way in which the above probability changes as the number of observation increases, suppose p was actually the true proportion of A’s supporters in the
population, i.e. p = º A . Let s¤ (p) denote the probability with which an expected
utility maximizer voter should choose opinion a; i.e. s¤ (p) = 0 for p < 1=2 , s¤ (p) = 1
for p > 1=2 and, conventionally, set s¤ (p) = 1=2 for p = 1=2. Then numeric computations show that, for the number of observations becoming very large and for each
given p = p, limn!1 Pr[a j n; p] = s¤ (p). In other words, for n large, Pr[a j n; p]
is essentially described by s¤ (p), although it remains di¤erentiable, since only in the
1
limit, for n ! 1, its image is restricted to the values 0; ; 1. However, for small
2
values of n, the probability with which the voter would formulate opinion a can be
9
substantially di¤erent from what (s)he would do, had (s)he perfect information about
the aggregate. This di¤erence, namely j Pr[a j n; p] ¡ s¤ (p) j, is a bias due to the fact
that information is limited.
We shall be interested in formalizing the process of opinion formation for very
small values of n. Intuitively, if a voter observes only a few observations and draws
inference on the aggregate, than such inference will only be partially correct. As priors
are uninformative, the voter’s opinion formation process will be strongly in‡uenced
by what (s)he observes. Such in‡uence is actually so strong to resemble pure imitative
behaviour, in the sense that a voter who goes through the above reasoning de facto
behaves in a way that reproduces the frequencies that (s)he observes. Speci…cally, let
v(p) = p for p 2 [0; 1] be the function that describes the very simple behaviour of a
voter who observes a randomly drawn observation from a binomial Bi(p) and imitates
such observation. Then for n · 4 the error we incur by approximating Pr[a j n; p]
with v(p) is bounded above by (2)n+1 . The next picture plots the two functions for
n = 4.
Uninformed Voter and Linear Voter: Probability of choosing a, as a function of p.
Although we shall not make use of this approximation in what follows, we consider
it to be an insightful relation between purely imitative behaviour on the one hand and
uninformed behaviour on the other. The above considerations provide the motivation
for the behavioral speci…cations we introduce here and to which we shall refer later.
De…nition 1 (Uninformed voter) Given n (number of observations being sampled) and p (parameter of the binomial distribution from which observations are
10
drawn) an uninformed voter chooses opinion a with probability:
Z1
n µ ¶
X
n
(n
+
1)!
z r (1¡z)n¡r dz
Pr U [a j n; p] =
pr (1¡p)n¡r
r
r!(n ¡ r)!
r=0
1 2
p 2 f0; ; :::; 1g
n n
1=2
(3)
De…nition 2 (Linear voter) Given n (number of observations being sampled) and
p (parameter of the binomial distribution from which observations are drawn) a linear
voter chooses opinion a with probability:
1 2
p 2 f0; ; :::; 1g
n n
Pr L [a j n; p] = p
(4)
Both speci…cations involve a process of sampling of n opinions among other voters:
in the …rst, an opinion is formed by updating priors on the basis of the observations,
in the second (a voter has no prior to update and) the opinion that is observed is
blindly imitated. Before we proceed, we stress that a key di¤erence between the two
speci…cations is the fact that for p = f0; 1g the two processes are the most di¤erent:
the linear voter can only imitate opinions that are observed, while the uninformed
voter who observes all identical opinions can still not imitate what (s)he sees on the
ground of the uniform prior.
2.2
Public Opinion Formation
As anticipated, we are going to model a dynamic process of public opinion formation,
in which we take the behavioral speci…cations introduced in the previous Subsection
as primitives. In order to account for di¤erent plausible information structures and
information transmission among agents, we will analyze di¤erent speci…cations of the
model.
The general notation of the model we study has individual x 2 S choosing opin-
ion ´(x) 2 fi = 0; 1g, where the set of opinion has been re-labeled for notational
convenience and opinion a is now re-labeled as 1. A con…guration of opinions in the
population will be denoted by ´ 2 f0; 1gS . We model the dynamics of the process
where at each point in time at most one individual changes opinion. To this aim,
we assume that each individual may choose a new opinion at a random exponential
time, with mean one. Whenever individual x is to form a new opinion, (s)he will do
11
so according to De…nition ?? or ??, according to the model we are studying. Within
the same model, all individuals form their opinion in exactly the same way.
We shall endogeneize the parameter p (that, we recall from the previous Section,
is the parameter that determines the probability of observing opinion 1) as pt (x; ´) ´
p(x; ´ t ), meaning that such probability depends on the agent’s identity, as well as
on the current con…guration of opinions in the population, but is homogeneous over
time.
The speci…cation of the model provides each agent with a spatial location on a
d-dimensional lattice Z d , and postulates that (s)he can only observe the opinions
adopted within the set of agents that live in her/his vicinity. Formally, we take
S = Z d and de…ne the set of x’s nearest neighbours as fy :k y ¡ x k= 1g, i.e. the
set of 2d agents who live at Euclidean distance one from agent x. We assume that
each voter is equally likely to observe any of the opinions adopted among her nearest
P
neighbors. As a result p(x; ´) = (2d)¡1 fy:ky¡xk=1g ´(y).
In general, we shall denote by ´ t the process at time t (´ t is clearly an element
of the state space f0; 1gS ) and we are interested in characterizing the time evolution
of the stochastic process ´t and its asymptotic properties as t ! 1. We denote any
probability distribution over the state space by ¹t , and the initial distribution by
¹0 . A degenerate probability distributions that has pointmass on the con…gurations
where all individuals adopt exactly the same opinion i (that is con…guration ´i where
¹
¹
´(x) = i for all x in S) is denoted by ¹i : Given ¹0 ; we let L(´ t 0 ) be the law of ´t 0 ,
¹
¹
¹
and we write limt!1 L(´t 0 ) = L(´ 10 ) to mean that L(´t 0 ) is weakly convergent. We
also denote by I the set of invariant measures for ´ t and Ie ½ I the set its extreme
points. We shall de…ne the process ´t to be ergodic if and only if I is a singleton; in
this case the above limit will not depend on the initial condition, in the sense that
¹
limt!1 L(´ t 0 ) = L(´ 1 ) for any ¹0 .
The following De…nitions summarize the details of the processes of public opinion
formation that we study.
De…nition 3 (Uninformed voters) Consider a population of S voters. At a random exponential time t, with mean one, voter x 2 S chooses opinion 1 with prob-
ability Pr U [1 j n; p(x; ´ t )] (given by equation (??)), where n < 1 and p(x; ´) 2
12
1 2
f0; ; :::; 1g: Then ´ U
t (n) de…nes the process of aggregate opinion formation for a
n n
population of uninformed voters, when sampling n observations.
De…nition 4 (Linear voters) Consider a population of S voters. At a random
exponential time t, with mean one, voter x 2 S chooses opinion 1 with probability
Pr L [1 j n; p(x; ´t )] = p(x; ´t ) (given by equation (??)), where n < 1 and p(x; ´) 2
1 2
f0; ; :::; 1g. Then ´L
t de…nes the process of aggregate opinion formation for a
n n
population of linear voters.
Before we proceed to state the main results of the paper, it is worth noticing that
some feature of the processes we shall analyze are relatively intuitive. First, since
whenever called to form an opinion, a voter samples current observations, and on the
basis of these (s)he decides, the aggregate process clearly satis…es some Markovian
properties. Second, as pointed out earlier, the uninformed voter who samples …nitely
many observations can choose each of the opinions with strictly positive probability.
This suggests that this process may be ergodic, in that all possible con…gurations
of opinions in the population could be visited in…nitely often by the process. In
this case a question of interest is the explicit characterization of its limit behavior.
This feature is clearly not shared by the linear voter process, as only opinions that
are observed can be adopted with positive probability. It should be clear from the
speci…cation that, for this latter process, the set of probability measures that have
pointmass one on state ´ i where everybody adopts opinion i, f¹i i 2 f0; 1gg, are
invariant. Although the lack of ergodicity does not allow for unique predictions as
for the limit behavior of this process, it is interesting to analyze what happens along
the dynamics of the process, in terms of the process of cluster formation.
3
Main Results
We are now ready to state the main results of the paper. The Theorems that follow
study the asymptotic behaviour of the two processes described in the previous Sections. The results are obtained for a speci…c characterization of the set of all voters,
that are located on a one-dimensional lattice and only observe a sample of opinions
from their nearest neighbors. This speci…cation is chosen mostly for convenience,
13
as it allows for an intuitive characterization of the space-time analysis we address
towards the end of this Section. Further comments on this particular issue are given
in a Remark in the Appendix.
Theorem 1 shows that when voters are uninformed, the process of public opinion formation is ergodic, in that it admits a unique limit distribution that is fully
characterized.
Theorem 1 Consider ´ U
t as in De…nition ?? and suppose that
X
1
´ t (y)
p(x; ´ t ) =
2
fy:ky¡xk=1g
For S = Z 1 , then:
1. for any n < 1, ´U
t (n) is ergodic (in the sense that I is a singleton) and
p
¹
2. for any initial ¹0 , L(´ t 0 ) ! L(´1 ) ´ ¹1 ; given by:
X
X
¹¾1 (´) = K exp[
¾(2´(x) ¡ 1)(2´(y) ¡ 1)]
where K is such that
Proof. See Appendix.
P
´
(5)
x fy:ky¡xk=1g
¹1 (´) = 1 and ¾ =
1
4
log(2n+1 ¡ 1) < 1.
The above Theorem provides a characterization of the limit behaviour of the
dynamics ´U
t : no matter where the process starts, the probability with which each
con…guration could be observed asymptotically is given by the above limiting distribution. As the latter has full support, each of the possible con…gurations of opinions
in the population can be observed in the limit. However, it is clear from the above
formulation that some con…gurations are more likely to be observed than others. In
particular, as the sum of which in the square brackets of (??) is taken over all couples
of nearest neighbours, and as the addendum is equal to one if and only if ´(x) = ´(y),
the two con…gurations which are more likely to be observed are those where every
voter chooses exactly the same opinion, i.e. ´ 0 and ´1 . Since (??) is continuous in
the parameter7 ¾, this proves the next Corollary.
Corollary 1 Under the assumptions of Theorem ??,
for all ¾
¹¾1 (´0 )
=1
¹¾1 (´1 )
and
14
¹¾1 (´)
=0
¾!1 ¹¾
1 (´ i )
lim
where f´i ; i 2 f0; 1gg with ´0 ´ f´ 2 f0; 1gS : ´(x) = 0
´(x) = 1
8x 2 Sg and ´ 2
= f´i ; i 2 f0; 1gg.
8x 2 Sg, ´ 1 ´ f´ 2 f0; 1gS :
The interpretation of the above Corollary in our model is the following. Since
¾=
1
4
log(2n+1 ¡ 1), taking a limit for ¾ ! 1 means studying what happens when
an uninformed voter samples an in…nite number of observations (n). Observations
are opinions randomly gathered in the neighbourhood and the interpretation we have
in mind is that each of these (even though they may come from the same neighbour)
provides the voter with some further information about the state of the system at the
time at which (s)he forms an opinion. In other words, a voter who repeatedly talks
to the same neighbours, updates her beliefs at each time. Ergodicity breaks down
only in the limit, as the transition probabilities of which in De…nition ?? become
discontinuous in the parameter p, as n grows to in…nity. It is clear that Corollary ??
only relies on a comparative static exercise over the limit distributions of a sequence
of identical processes, that di¤er only in the parameter ¾ and, as such, it does not
provide a full understanding of the dynamics at any …nite point in time.
The message that the above result conveys is that, asymptotically, we are more
likely to observe a con…guration of homogeneous opinions in the population. Due to
the underlying symmetries that the process satis…es, this is perhaps not surprising.
Given these asymptotics (and given that elections typically take place at a …nite,
rather than in…nite, time), what we would like to know in more detail what happens
along the dynamics of the process. Loosely speaking, we would like to know to what
extent the logic we followed in Theorem ?? (where we …rst looked at the asymptotics
for t ! 1, and then at the limit for ¾ ! 1) can be reversed by …rst looking at
a process where a voter knows exactly what the opinions in her neighbourhood is
(i.e. in the limit for ¾ ! 1) and second at what happens along the dynamics of
this process (i.e. asymptotically for t ! 1). By this doing we can gather some
further understanding of the way the process evolves, when the behaviour of voters
is not driven by lack of information about the current con…guration of opinions in
the neighbourhood. The Theorem that follows establishes that this line can be pursued and that the resulting process is exactly the process ´L
t , for which asymptotic
behaviour is characterized.
15
1
Theorem 2 Consider ´ U
t (¾) and let S = Z : Then:
L
1. for any given t, ´U
t (1) corresponds to ´ t (as in De…nition ??).
2. Consider ´ L
t , and for 0 · µ · 1, let ¹µ be the product measure with density µ,
i.e. ¹µ f´(x) = 1g = µ for all x 2 S. Then:
¹
Ie = f¹0 ; ¹1 g and L(´t µ ) ! (1 ¡ µ)¹0 + µ¹1
Proof. See Appendix.
The above Theorem shows that the process ´L
t can be interpreted as a limiting case
of the process ´U
t and that this latter process admits no extreme invariant measures
other than f¹0 ; ¹1 g. Part (2.) is relevant for our purposes because it shows that
U
the extreme invariant measures of ´L
t are exactly those to which ´ t (¾) collapses for
¾ ! 1. If this was not the case the invariant measures of ´ L
t could be of a di¤erent
nature than ´U
t (1); they could for example identify con…gurations where di¤erent
opinions coexist in equilibrium, and this (from Corollary ??) would not be consistent
with the process ´ U
t .
The second reason why the above result is important is that it shows that along
the dynamics, the process shows consensus, in that if we look at any possible couple
of voters, x and y in S, the probability that they choose di¤erent opinions approaches
zero asymptotically:
lim Pr[´t (x) 6= ´t (y)] = 0 for all x and y in S
t!1
Clearly, for any 0 < µ < 1, each single voter may change her or his opinion
in…nitely many times (as limt!1 ´t (x) does not necessarily exist). However, as a
result of the above considerations, the observed frequencies of individuals choosing
the same opinion grows over time. Our aim is now to characterize more in detail how
this occurs.
As anticipated, we intend to study the properties of the dynamics of our process
for any …nite t. As our process is de…ned in the two dimensions of time and space,
we shall …nd it useful to relate these two dimensions in a space-time analysis. In
particular, we aim at characterizing a clustering process, by relying on the local
speci…cation of the model. With the term “cluster” we mean a connected group
16
of individuals holding the same opinion, that is the length of a segment with all
connected individuals of the same opinion. In order to see how the size of a cluster
increases with time, we shall later express the length of a cluster as a function of t.
Formally, given a con…guration, ´, we de…ne a cluster as the connected components
of fx : ´(x) = 0g or fx : ´(x) = 1g and the mean cluster size of ´ (around the origin)
as:
2l
l!1 ‘number of clusters of ´ in [¡l; l]’
C(´) = lim
whenever this limit exists.
The Theorem that follows states that, if the initial distribution is a product
p
measure, and if we re-scale the length of a cluster as l = t, then the mean cluster
size around the origin converges in distribution, as t ! 1. Its limit depends on the
initial distribution and it is possible to provide bounds, expressed as a function of
this probability.
Theorem 3 (Bramson and Gri¤eath (1980)) Consider ´ L
t (as in De…nition ??)
and for 0 < µ < 1, let ¹µ be the product measure8 with density µ, i.e. ¹µ f´(x) = 1g =
µ for all x 2 S = Z 1 . Then, for any initial distribution ¹µ :
p
¼
µ
1
2µ(1 ¡ µ)
¶
· lim E[
t!1
µ 2
¶
C ¹µ (´t )
µ + (1 ¡ µ)2 p
p
¼
]·2
µ(1 ¡ µ)
t
Proof. Theorem 7 in Bramson and Gri¤eath (1980), p. 211.
An immediate corollary of the above Theorem is the fact that, as t gets large,
the largest segment containing all individuals choosing the same opinion, has side of
p
probability order t. For t very large, such cluster tends to be almost stationary,
in the sense that the rate at which it changes is slower than the rate at which time
changes9 .
The Theorem provides a numerical lower and an upper bound for the expected
mean cluster size. To interpret this estimate, consider a process that starts with an
initial distribution where each voter chooses opinion 1 with probability, say, µ = 12 :
As choices are initially independent, clearly, at time zero, the probability of observing
a cluster of k = 100 voters with the same opinion is 2¡100 . As the process evolves,
however, choices show a certain amount of spatial correlation. For t ! 1 the mean
17
p
p
cluster size, re-scaled by t, will converge to a limit that lies between 2 ¼ = 3: 5449
p
and 4 ¼ = 7: 0898. Hence a cluster of k = 100 voters could be approximately
observed as early as after t = 198: 94
10 ,
and is on average not going to vary until
t = 795: 78, as the …gure that follows illustrates:
Limiting mean cluster size for µ = 12 :
In other words, in order to observe the cluster size to double (say from k = 100
to k = 200 in the above picture), the process needs to go through four times as many
periods (say from t » 200 to t » 800).
Simple calculus shows that the lower and the upper bound of the (limiting) mean
cluster size are convex in µ and symmetric around µ = 12 . Hence for µ 6=
of a given mean size is likely to be observed earlier than if µ was
1
2
1
2
a cluster
and is likely
to ‘persist’ for a relatively longer spell of time. Hence, conditional on a candidate
winning the elections, his or her support in terms of absolute number of votes grows
p
at rate t. The higher is µ, the lower is the number of time periods that are necessary
to achieve a given minimum expected cluster size of votes in her or his favour, and
the longer is the spell of time within which his or her electoral support is going to
remain almost stationary.
4
Insights for further research
As the dynamics we studied are speci…ed over time and over space, natural questions
to be addressed relate to the optimal spatial allocation of funding in an electoral
campaign (i.e. among di¤erent districts or di¤erent states), as well as to the optimal
timing of such allocation (i.e. between the time when the elections are called and
18
the time just before the elections are actually held). Although a formal treatment of
these interesting questions warrants future research, in what follows we elaborate on
the insights that the model we studied in this paper provides.
The …rst thing that all speci…cations of our model show is that the spatial distribution of votes matters in the long run, as well as in the short run. In particular,
simply by looking at the limit distribution for the ergodic process generated by the
dynamics of the uninformed voters model, as in Theorem 1, it is easy to see that
the limit probability of each con…guration depends on the opinions chosen in its connected components, and not on the frequency with which opinions are adopted in
the population. For example, consider two con…gurations, ´ A and ´ B , identical at all
sites apart from the sites fx ¡ 2; x ¡ 1; x; x + 1g which are as follows:
´A : :::: ´(x ¡ 2) = 1 ´(x ¡ 1) = 0 ´(x) = 1 ´(x + 1) = 0 ::::
´B : :::: ´(x ¡ 2) = 1 ´(x ¡ 1) = 1 ´(x) = 0 ´(x + 1) = 0 ::::
From Theorem 1 we infer that the limit probabilities of these con…gurations (where
the frequencies of 1s is exactly the same) are respectively:
¹¾1 (´A ) / exp[¡6¾]
¹¾1 (´ B ) / exp[2¾]
Con…guration ´B is given higher probability, as more coordinates agree with their
neighbouring coordinates. These considerations clearly relate to the long-run distribution of the process, but the insight applies to the short-run, as can be seen by
looking at the dynamics of the speci…cation of the model in terms of linear voters, to
which we focus next.
Much of the descriptive and normative literature on elections in political science
identi…es at least two alternative basic rules that a candidate may follow when deciding where to allocate resources (in terms of money, as well as time spent campaigning)
among di¤erent constituencies or states. The …rst posits that a candidate should allocate campaign resources roughly in proportion to the electoral votes of each state
(Brams and Davis (1974)). The second suggests that candidates should mostly be
concerned with the likelihood that resources can swing a state from one candidate to
another, and by this advocates a competitive allocation of resources to be directed
to the ‘marginal’ states (Colantoni et al. (1974)). With some heroic simpli…cations,
19
we can translate these two alternatives into the set-up of our model, by asking the
following question: suppose a candidate had the possibility to buy one vote (i.e. to
buy the support of one voter), would (s)he rather do so within a cluster of voters
who support the other candidate, or exactly at the border of a cluster? It turns
out that our model suggests that the best alternative is this latter possibility. To
see this, consider the following con…guration, ´, that has a border at x = 0, in that
´(x ¡ 1) 6= ´(x):
:::: ´(x ¡ 2) = 1 ´(x ¡ 1) = 1 ´(x) = 0 ´(x + 1) = 0 ´(x + 2) = 0 ::::
Suppose, for simplicity, that the process is started deterministically at con…guration ´. In this case the duality equation (??) (see the proof of Theorem 2 in the
Appendix) states that the probability that starting from con…guration ´, the voter
P
at site x supports candidate 1 is: E ´ ´t (x) = y pt (x; y)´(y), which, applied to the
subset fx ¡ 1; x; x + 1g; becomes:
E ´ [´t (x¡1)+´t (x)+´ t (x+1)] =
X
y
X
X
pt (x¡1; y)´(y)+
pt (x; y)´(y)+
pt (x+1; y)´(y)
y
y
The above probabilities are given explicitly in equation (??), and it is not di¢cult
to see that, for any …nite t, since pt (x; x + j) = pt (x; x ¡ j) for any j ¸ 1 and since
p(0) (x; x + 1) = p(0) (x; x ¡ 1) = 12 :
1
¸ pt (x; x + 1) ¡ pt (x; x + j) > 0
2
8j > 1
formalizing the fact that a voter’s opinion is more strongly a¤ected by the opinions
held in the neighbourhood than by opinions held further away.
If we take into account of this fact, and we denote pt (x; x+1) as p, we can re-write
the above equation as:
E ´ [´t (x ¡ 1) + ´t (x) + ´t (x + 1)] ¼ p[´(x ¡ 2) + ´(x ¡ 1) + 2´(x) + ´(x + 1) + ´(x + 2)]
= p[1 + 1 + 2´(x) + ´(x + 1) + ´(x + 2)]
Hence, by buying the vote of voter x, candidate 1 increases the probability that
at time t voters in fx ¡ 1; x; x + 1g support her or him by twice as much as (s)he
would do by buying the vote of voter x + 1 or voter x + 2. This is because by moving
20
the border of a cluster by one voter, the candidate guarantees stability of the area
inside the cluster, that being inward looking is not so exposed to sudden swings in
opinions.
A further insight that the model provides relates to the optimal timing of resource
allocations in an electoral campaign. As we showed before, in the linear voter model
the process is path-dependent, as its long run behaviour depends crucially on the
initial distribution. This determines the basins of attraction of the two limit distributions (that, we recall, show consensus), as well as the lower and upper bound of
the expected minimum cluster size. Hence the model suggests that what happens
at the very beginning of an electoral campaign has a very strong e¤ect on its later
developments, and raises the incentive for a candidate to invest campaign resources
on whatever is deemed to have any power to a¤ect the initial distribution. Along the
dynamics, clusters emerge and are almost stationary when viewed locally. Clearly, as
the model is non-ergodic, there is no guarantee that once a minimum cluster size is
reached, electoral support for a candidate will continue to grow unboundedly. Hence,
if a candidate could gather some information about the current distribution of potential votes (for example through an electoral poll) and if this was favourable to her
or him, then delaying the date of the elections could have a detrimental e¤ect on the
p
outcome. As clusters grow at rate t the model also seems to suggest that a linear
allocation of funding over time during an electoral campaign might be sub-optimal,
as the returns in terms of electoral support are decreasing over time11 .
5
Concluding remarks
This paper studied a dynamic model of pre-electoral public opinion formation. We
treated public opinion as a continuously changing process and we analyzed the emergence of interactive patterns of behavior.
The model involves a countable population of individuals that repeatedly choose
to support one of two candidates. Each individual has a well de…ned preference
structure over the …nal electoral outcome that formalizes an incentive to conform to
the opinion held by a perceived majority. In the …rst speci…cation of the model agents
update their beliefs over the current distribution of opinions by sampling a number
21
of observations within their neighbours. In the second, voters simply follow a linear
rule, by choosing opinion 1 with probability ® if ®% of their neighbours hold opinion
1 (and viceversa).
We analyzed the dynamics of the public opinion process by addressing two related
questions. The …rst relates to the asymptotics of the process. For certain speci…cations we showed that the process is ergodic, while for others the process admits two
extreme invariant measures, where one candidate is supported by the whole population. The second question explicitly focused on the dynamics itself, by pursuing a
space-time analysis. It turned out that the process displays the feature that, after
long time spans, a sequence of states occur which, when viewed locally, remain almost
stationary and are characterized by large clusters of individuals of the same opinion.
Finally, we provided some heuristic considerations on the implications that these
…ndings could have within a more general model that allows for strategic behaviour
on the part of the two candidates.
22
Appendix
Proof of Theorem 1
(1) (ergodicity)
We interpret the process ´U as a system of interactive, nearest neighbours, particles on a one-dimensional lattice, Z 1 . We …rst show that the process is attractive
(or monotonic) in that coordinates tend to agree with neighbouring coordinates. It
is known (see, for example, Liggett (1985), Theorem 3.14, p.152) that a su¢cient
condition for an attractive system with a countable state-space to be ergodic, is that
the transition probabilities that generate the process be strictly positive. This is the
logic we follow.
1
We introduce the following partial order on f0; 1gZ . We say that, for ´; ³ 2
1
f0; 1gZ , ´ · ³ if ´(x) · ³(x) for all x 2 Z 1 . Then a process is de…ned to be
attractive if, whenever ´ · ³ :
c(x; ´) · c(x; ³) if ´(x) = ³(x) = 0
c(x; ´) ¸ c(x; ³) if ´(x) = ³(x) = 1
where c(x; ¢) are the ‡ip rates that generate the dynamics (i.e. c(x; ¢) is the probability
with which coordinate x ‡ips, in state ¢).
In order to check this condition, and for later purposes, we re-write the transition
probabilities of which in equation (??) by substituting ¾ =
Pr U [1
1
4
log(2n+1 ¡ 1):
n; p(x; ´)] ´ Pr U [1 j ¾; p(x; ´)] =
(6)
1
=
(7)
1 + exp[¡4¾(2p(x; ´) ¡ 1)]
P
where we recall p(x; ´) = 12 y:ky¡xk=1 ´(y) and takes values in f0; 1=2; 1g. For
j
example, if p(x; ´) = 0 the above equation states that the probability that opinion 1
is chosen is given by [1 + exp[4¾]]¡1 = [1 + exp[log[2n+1 ¡ 1]]]¡1 = (2n+1 )¡1 .
Hence, the ‡ip rates can be written as:
1
1 + exp[¡4¾(2p(x; ´) ¡ 1)]
1
Pr U [0 j ¾; p(x; ´); ´(x) = 1] =
1 + exp[+4¾(2p(x; ´) ¡ 1)]
Pr U [1 j ¾; p(x; ´); ´(x) = 0] =
23
As clearly p(x; ´) · p(x; ³) whenever ´ · ³, our process is attractive. It is also
clear from equation (??) with n < 1, or equivalently from the above speci…cations,
for ¾ < 1, that transition probabilities are strictly positive. Hence I is a singleton.
(2) (characterization of the limit distribution)
In order to characterize the unique invariant measure of the process, we establish
a relation between the process ´U
t and a class of stochastic processes known as Ising
models (Liggett (1985), Chapter IV provides details on Ising models).
It is easy to see that the transition probabilities of which in equation (??) correspond exactly to the ‡ip rates of a stochastic Ising model, with nearest neighbour
interactions, relative to the following potential:
8
< ¾ if R = fx; yg and y :k y ¡ x k= 1
JR =
: 0 otherwise
(8)
Hence the unique limit distribution of the process is the Gibbs state corresponding
to the above potential:
X
¹¾1 (´) = K exp[
P
where K = f
´
X
x fy:jy¡xj=1g
¾(2´(x) ¡ 1)(2´(y) ¡ 1)]
P P
exp[ x fy:jy¡xj=1g ¾(2´(x) ¡ 1)(2´(y) ¡ 1)]g¡1 (see for example
equation (1.4) on p. 180 in Liggett (1985)).
To see this, it su¢ces to notice that the above measure is reversible, in that:
Pr U [1 j ¾; p(x; ´); ´(x) = 0]¹¾1 (´ x=0 ) = Pr U [0 j ¾; p(x; ´); ´(x) = 1]¹¾1 (´x=1 )
where the two con…gurations ´ x=0 and ´x=1 di¤er only in the coordinate x (i.e.
´x=0 (x) = 0; ´ x=1 (x) = 1 and ´x=0 (y) = ´ x=1 (y) for all y 6= x):
¹¾1 (´x=1 )
¹¾1 (´x=0 )
= exp[2¾
X
(2´(y) ¡ 1)]
fy:ky¡xk=1g
=
1 + exp[¡2¾
¢[
=
1 + exp[2¾
P
1
fy:ky¡xk=1g (2´(y) ¡ 1)]
P
1
fy:ky¡xk=1g (2´(y) ¡ 1)]
Pr U [1 j ¾; p(x; ´); ´(x) = 0]
Pr U [0 j ¾; p(x; ´); ´(x) = 1]
24
¢
]¡1
Clearly any reversible measure is also an invariant measure (i.e. ¹¾1 2 I ). From
part (1) we know that the process is ergodic, i.e. I is a singleton. Hence the assert
follows.
Proof of Theorem ??
1. Recall that the process ´ U
t (1) is de…ned by the transition probabilities of
which in (??). Hence, we only need to show that lim¾!1 Pr U [1 j ¾; p(x; ´)] = p(x; ´)
for p(x; ´) 2 f0; 12 ; 1g. This is clear by looking at (??) and taking such limit, for any
given p(x; ´).
2. Clearly, for the process ´ L
t , I ¶ Ie ¶ f¹0 ; ¹1 g, as by simple inspection of the
1
transition probabilities that de…ne the process (namely p(x; ´) for p 2 f0; ; 1g) it
2
is clear that any state for which ´(x) = ´(y) for all x; y in S is stationary for the
process. Hence, the result relies on the proof that these are the only two extreme
invariant measures (i.e. Ie µ f¹0 ; ¹1 g), so that, as I is a convex set, any other
invariant measure is fully characterized. Furthermore, one needs to show that the
domains of attraction of each extreme invariant measure, depend on the stochastic
¹
initial condition given by the product measure ¹µ , as in L(´t µ ) ! (1 ¡ µ)¹0 + µ¹1 .
Results along these lines are well known in the statistical literature on the Voter’s
model in the case Z 1 and can be found in Liggett (1985), Section 1 and 3, Chapter
V or in Bramson and Gri¤eath (1980). As the logic of the proofs is interesting in its
own right, we sketch the proof in what follows.
The process ´ L
t (shortened to ´ t in what follows) can be studied in terms of its
dual process in terms of coalescing random walks. The duality relation transforms
questions about ´t in questions concerning the cardinality of the coalescing random
walk system.
We …rst show that such duality can be used, by checking the conditions of which
in equation. (4.3) (p. 158) in Liggett (1985). To this aim, note that the ‡ip rates for
the process ´ L
t can be written as:
c(x; ´) = ´(x) + p(x; ´)(1 ¡ 2´(x))
= (1 ¡ ´(x)) + (2´(x) ¡ 1)
25
X
fy:ky¡xk=1g
1
(1 ¡ ´(y))
2
as p(x; ´) =
P
1
fy:ky¡xk=1g 2 ´(y).
These coincide with equation. (4.3) (p. 158) in
Liggett (1985), once we take c(x) = 1, A = fyg and p(x; A) = p(x; y) =
1
2
if y :k
y ¡ x k= 1 and zero otherwise.
The dual process is a system of countably many continuous time, symmetric
random walks that jump after an exponential mean-1 holding time, with probabilities
p(x; x + 1) = p(x; x ¡ 1) = 12 . Whenever two random walks meet (i.e. if one jumps
to a site that is already occupied), then they coalesce, i.e. they merge into one. In
particular, any such random walk de…nes a continuous time Markov chain, X(t), with
transition probabilities:
pt (x; y) = e¡t
1 n
X
t
n!
n=0
p(n) (x; y)
(9)
where p(n) (x; y) are the n-step transition probabilities associated with p(x; y). Any
system of …nitely many independent copies of X(t); where any two merge whenever
they meet, de…nes a system of …nitely many coalescing Markov chains over the state
space of all …nite subsets of S.
We denote by At the system of coalescing random walks at time t, that started
at time zero in the …nite subset A ½ S. For any such subset A, let:
gt (A) = Pr A [j At j<j A j for some t ¸ 0]
where j ¢ j denotes the cardinality of a set. This represents a measure of how far
apart the single processes are. Clearly, for any t, gt (A) = 0 when j A j= 1, as a single
recurrent random walk is never going to die. If j A j= 2, gt (A) !t!1 1; meaning
that two recurrent random walks will tend to meet and coalesce, as time grows, and
possibly only asymptotically. In order to shorten an otherwise very long proof, we
shall however assume that gt¤ (A) = 1 when j A j= 2 for some t¤ < 1.
Let A = fx 2 S : ´(x) = 1 for all x 2 Ag and, for ¹ being a probability measure
on f0; 1gS , let ¹(A) = ¹f´ : ´(x) = 1 for all x 2 Ag. Then the duality equation can
be stated as follows (see equation. 1.7, p. 230 in Liggett (1985)):
¹t (A) = E A ¹(At )
(10)
where ¹t (A) is the probability that the process ´t has ´ t (x) = 1 for all x 2 A and
E A ¹(A
t)
time t.
is the probability that j At j random walks, started at A, are still alive at
26
By using this duality relation, we now show that, given a product measure ¹µ ,
¹
L(´t µ ) ! (1 ¡ µ)¹0 + µ¹1 .
To characterize the basins of attraction of f´0 ; ´1 g, suppose the process ´ t is
started (stochastically) with product measure ¹µ . If ¿ is the …rst time that j At j= 1
(which is …nite with probability one by our assumption that gt¤ (A) = 1 when j A j= 2)
the duality equation (??) implies that:
lim E A ¹(At ) = E A [ lim E A¿ ¹(At )]
t!1
t!1
Applying this again to A = fxg we obtain:
lim
t!1
X
y
pt (x; y)¹(fyg) = µ for all x 2 S
But, by part (b) of Theorem 1.9 in Liggett (1985) (p. 231), this is a necessary
¹
and su¢cient condition for L(´ t µ ) ! (1 ¡ µ)¹0 + µ¹1 to be true. Hence the assert
follows.
Remark
The speci…cation of the model we use throughout the paper allows for a countable
set of individuals located on a one-dimensional lattice. One may wonder whether the
results are peculiar to this characterization, other things being equal.
Most of the results would also hold for a …nite (hence countable) set of individuals.
In this case the dynamics of the uninformed voter model and that of the linear voter
model would de…ne a continuous time Markov chain over a …nite state-space. The
analog of Theorem 1 (and its Corollary) would hold in any dimension, on the grounds
that the chain would be ergodic (whereas for a countable state-space, it is known
that in a d-dimensional setting with d ¸ 2 the process may admit multiple invariant
distributions). As for the analog of Theorem 2, independently of the dimension, the
process ´L
t would converge with probability one to f´ i ; i 2 f0; 1gg, as the chain would
be absorbing, with these con…gurations as the only absorbing states. As a result,
for any given initial condition, the process would get trapped (in …nite time) in a
con…guration that shows consensus. Unlike in the case of a countable population,
each voter could change her or his opinion only …nitely many times. We conjecture
that also the domains of attraction of the two stationary measures, ¹0 and ¹1 , would
27
be exactly the same. However, the technique used in the proof we provide, as well
as that used to proof Theorem 3 (Bramson and Gri¤eath (1980)) heavily relies on
the duality in terms of coalescing random walks and it is known that these behave
di¤erently for d · 2 and for d > 2. In particular, as a …nite system will be trapped
with probability one, the analysis of the clustering process could not be carried out.
An interesting question to address in this case is the study of the way absorption times
vary, for the number of individuals growing large (as in Cox (1989)). In Corradi and
Ianni (1998b) we study the details of the clustering process in a 2-dimensional model
related to those we analyzed in this paper.
28
Notes
1
The wording is by I. Crespi, the former president of the American Association for Public Opinion
Research.
2
In Corradi and Ianni (1998a) we further investigate the relation between this clustering process
and stationary co-existence of di¤erent opinions in a model analog to the one we study in this paper,
for di¤erent speci…cations of the dynamics.
3
As we shall discuss later, the focus on a countable (not necessarily …nite) population of individuals
is motivated by the fact that we want to analyze explicitly the process of cluster formation. To this
aim, we shall assume that the set of individuals is located on a one dimensional lattice. Consistently,
the frequency of opinions in the population is de…ned as the limit of its natural restriction to [¡l; l],
as l ! 1. For the purposes of the exposition, we assume that this limit exists when describing
the process of Private Opinion Formation. As it will become clear, such assumption will be trivially
satis…ed by our process of Public Opinion Formation, as interactions will only have …nite range.
4
Riker and Ordershook (1968) estimated such probability for the USA as being 10¡8 :
5
An interesting philosophical discussion of rational voting decisions is provided in Meehl (1977).
6
This assumption is only introduced for technical convenience. It is consistent with the idea of
symmetric (equilibrium) behaviour, on the part of the two candidates, if side-payments where decided
strategically. As we formally show in Corradi and Ianni (1998a), the dynamics of the public opinion
process in the case where the side-payments are asymmetric are far less articulate than otherwise.
7
More precisely, it varies upper hemicontinuously with ¾ in the weak convergence topology.
8
The result is actually true for any n-fold mixing measure as de…ned in Bramson and Gri¤eath
(1980).
9
In Ellison (1993) the author studies the rates of convergence of best-reply dynamics for an
underlying coordination game, repeatedly played by couples of players drawn at random from a
…nite population. Our model di¤ers from the cited paper in a number of respects. First, the
speci…cation of the dynamics that Ellison (1993) studies is perturbed by mistakes (that take the
form of a binomial distribution that assigns small, though strictly positive, probability, uncorrelated
across players and over time, to actions that are not best-replies to the current con…guration of play).
This is substantially di¤erent from the way we model the individual process of opinion formation
(that in the speci…cation in terms of uninformed voters could be motivated in terms of mistakes that
do depend on expected payo¤s). Second, the dynamics of Ellison’s (1993) are de…ned over a …nite
state-space and modeled as …nite, discrete time, regular Markov chains, whereas our dynamics de…ne
a Markovian process over a countable state-space, that is ergodic if voters are ‘uninformed’, but is
path-dependent if voters are ‘linear’. Lastly, the cited paper compares the speed of convergence of
29
transition probabilities to their limit values, in a model with local interaction and in an (analog)
model with global interaction. All speci…cations of our model rely on a local characterization of the
way in which interaction takes place.
10
Of course the quality of the approximation improves with t.
11
However, considerations of this sort require an explicit consideration of the strategic interaction
between the two candidates, which at present is not part of the model.
30
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